We consider the viscosity solution of a homogeneous Dirichlet problem for the eikonal equation in a bounded set Ω. We suppose that the Hamiltonian, H(x,p)=〈A(x)p,p〉−1, is strictly convex w.r.t. the variables pp and of class C1,1 w.r.t. the variables x. Then the solution of the Dirichlet problem admits an extension to a neighbourhood of Ω, such that u is still a viscosity solution of the eikonal equation if and only if ∂Ω satisfies an exterior sphere condition. The above result, in particular, provides a characterization of the boundary singularities and a regularity theorem (up to the boundary) for the solution of the eikonal equation.

On the extension of the solutions of Hamilton-Jacobi equations

ALBANO, PAOLO
2011

Abstract

We consider the viscosity solution of a homogeneous Dirichlet problem for the eikonal equation in a bounded set Ω. We suppose that the Hamiltonian, H(x,p)=〈A(x)p,p〉−1, is strictly convex w.r.t. the variables pp and of class C1,1 w.r.t. the variables x. Then the solution of the Dirichlet problem admits an extension to a neighbourhood of Ω, such that u is still a viscosity solution of the eikonal equation if and only if ∂Ω satisfies an exterior sphere condition. The above result, in particular, provides a characterization of the boundary singularities and a regularity theorem (up to the boundary) for the solution of the eikonal equation.
2011
Paolo Albano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/154667
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